My Traveller campaigns use the following material:
The Traveller Book
Book 5: High Guard
Supplement 2: Animal Encounters
A brief nod to two I am considering using:
Supplement 1: 1001 Characters
Supplement 4: Citizens of the Imperium
In the above, I use Book 5 not for its character generation sequence, but for its starship construction rules. After all, creating 190,000 ton starships isn't all that easy to do using the basic Starships rules, though you've managed to spark an idea in my head on how to do it without having to craft 37 million rules to do it.
Supplement 2 is there simply because I, like just about every other GM, couldn't stand making up the encounter tables.
Supplement 1 is useful simply because you never know when you might need an NPC. The Traveller Book includes a short list of NPCs, but there's only so far you can go with that.
Supplement 4 I'm vacillating back and forth on. The additional careers are good, but I tend to toss the non-military careers into the "Other". To my mind it gives the player the opportunity to create a good backstory for the character.
The core books will always be relevant, for without them, there would be no Traveller. And though the systems in some cases are aged, they still do the job. I won't get involved in the debates over Traveller with a task system or without. Both have their pros and cons but Andy Slack has already written up a quick and dirty task system for his Traveller Ultralite rules. That, coupled with the mad idea in my head for starships, might allow me to cut my game down to the core rules and one added page. And that would be great, because I like the portability the LBBs offer in place of The Traveller Book.